What number is 110% of 150?

Set up a proportion

#x# = the 110% of 150 the part
150 = the total parts y
110% = the percent part
100% = the total percent.

Using the principle of ratio I choose to set 100 as the equivalent of the original quantity of 150 and 110 as the equivalent of the new unknown quantity.

Let #x# be the new unknown quantity giving the ratio

#("New quantity")/("original quantity") =110/100=x/150#

Multiply both sided by 150

# ( 110 xx 150)/ 100 = ( x xx 150)/150#

This gives

# 16500/100 = x #

# 165 = x #

#color(blue)("The underlying principle of percentage")#

#color(red)("It just a fraction.")# However, it is a special fraction because the bottom number (denominator) is fixed at 100.

The symbol % behaves like a unit of measurement but one that is worth #xx1/100# including the multiply sign

So #110%# is the same as #110xx1/100 = 110/100#

The question states #110% color(red)(" of ") 150#

Translating the word 'of' into mathematics it becomes multiply

#color(green)(110color(white)("dd")color(blue)(%)color(white)("dd.")color(red)("of")color(white)("dd")150)#

#color(green)(110color(white)("dd")color(blue)(%)color(white)("dd.")color(red)(xx)color(white)("d")150)#

#color(green)(110color(blue)(obrace(xx1/100))color(white)("d")color(red)(xx)color(white)("d")150)#

Giving:

#110/100xx150 larr" Starting point"#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("A trick to calculating this in the head")#

#110/100 -> 100/100+10/100 = 1 +1/10#

#color(white)("d")#

#color(white)("d")1color(white)(".")xx150=150#
#1/10xx150= ul(color(white)(1)15larr" Add")#
#color(white)("dddddddddd")165#